Augmentations and Rulings of Legendrian Links in $\#^k(S^1\times S^2)$

TL;DR

This paper extends the concept of normal rulings to Legendrian links in connected sums of S^1 times S^2, establishing a correspondence with augmentations of the Chekanov-Eliashberg algebra and deriving implications for symplectic homology.

Contribution

It generalizes normal rulings to new 3-manifolds and links them to algebraic augmentations, providing new tools for symplectic topology.

Findings

Normal rulings are equivalent to augmentations over any field.

Even graded augmentations send t to -1 for Legendrian knots.

Results imply nonvanishing of symplectic homology in certain Weinstein 4-manifolds.

Abstract

Given a Legendrian link in $\#^k(S^1\times S^2)$, we extend the definition of a normal ruling from $J^1(S^1)$ given by Lavrov and Rutherford and show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over $\mathbb{Z}[t,t^{-1}]$ is equivalent to the existence of a normal ruling of the front diagram. For Legendrian knots, we also show that any even graded augmentation must send $t$ to $-1$. We use the correspondence to give nonvanishing results for the symplectic homology of certain Weinstein $4$-manifolds. We show a similar correspondence for the related case of Legendrian links in $J^1(S^1)$, the solid torus.